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FINITE ELEMENT VIBRATION ANALYSIS OF ROTATING TIMOSHENKO BEAMS
Journal of Sound and
FINITE ELEMENT VIBRATION ANALYSIS OF ROTATING TIMOSHENKO BEAMS S. S. RAO Department of Mechanical Engineering, ;niversity of Miami, Coral Gables, F¸ 33124-0624, ;.S.A. AND
R. S. GUPTA Department of Mechanical Engineering, Punjab Engineering College, Chandigarh 11, India (Received 12 January 2000, and in ,nal form 13 September 2000) The sti!ness and mass matrices of a rotating twisted and tapered beam element are derived. The angle of twist, breadth and depth are assumed to vary linearly along the length of beam. The e!ects of shear deformation and rotary inertia are also considered in deriving the elemental matrices. The "rst four natural frequencies and mode shapes in bending}bending mode are calculated for cantilever beams. The e!ects of twist, o!set, speed of rotation and variation of depth and breadth taper ratios are studied. 2001 Academic Press 1. INTRODUCTION
Generally, turbomachine blades are idealized as tapered cantilever beams. In order to re"ne the analysis the e!ects of pre-twist and rotation are also included. As the blades are short in some of the designs and may vibrate in higher-frequency ranges, the e!ects of shear deformation and rotary inertia may be of considerable magnitude. Various investigators in the "eld of turbine-blade vibrations have solved the di!erential equations of motion, by taking into account one or more of the above-mentioned e!ects. The analysis of tapered beams has been made by many investigators using di!erent techniques. Rao [1] used the Galerkin method, Housner and Keightley [2] applied the Myklestad procedure, Rao and Carnegie [3] used the "nite di!erences approach, Martin [4] adopted a perturbation technique and Mabie and Rogers [5] solved the di!erential equations using Bessel functions to "nd the natural frequencies of vibration of tapered cantilever beams. In analysing the pre-twisted beams, various approaches have been used by various investigators. Mendelson and Gendler [6] used the station functions, Rosard [7] applied the Myklestad method, Di Prima and Handleman [8] adopted the Rayleigh}Ritz procedure, Carnegie and Thomas [9] used the "nite di!erence techniques and Rao [10] considered the Galerkin method for the analysis of pre-twisted beams. The bending vibration of a twisted rotating beam was considered by Targo! [11]. The rotation e!ect in beam analysis has also been considered using di!erent formulation and solution procedures. Targo! applied the matrix method, Yntema [12] applied the Rayleigh}Ritz procedure, Isakson and Eisley [13] used the Rayleigh}Southwell procedure, Banerjee and Rao [14] applied the Galerkin procedure, Carnegie et al. [15] applied the "nite di!erence scheme, Krupka and Baumanis [16] used the Myklestad method and Rao 0022-460X/01/160103#22 $35.00/0
2001 Academic Press
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S. S. RAO AND R. S. GUPTA
and Carnegie [17] used the extended Holzer method to "nd the vibration characteristics of rotating cantilever beams. The "nite element technique has also been applied by many investigators, mostly for the vibration analysis of beams of uniform cross-section. All these investigations di!er from one another in the nodal degrees of freedom taken for deriving the elemental sti!ness and mass matrices. McCalley [18] derived the consistent mass and sti!ness matrices by selecting the total de#ection and total slope as nodal co-ordinates. Kapur [19] took bending de#ection, shear de#ection, bending slope and shear slope as nodal degrees of freedom and derived the elemental matrices of beams with linearly varying inertia. Carnegie et al. [20] analyzed uniform beams by taking few internal nodes in it. Nickel and Sector [21] used total de#ection, total slope and bending slope of the two nodes and the bending slope at the mid-point of the beam as the degrees of freedom to derive the elemental sti!ness and the mass matrices of order seven. Thomas and Abbas [22] analyzed uniform Timoshenko beams by taking total de#ection, total slope, bending slope and the derivative of the bending slope as nodal degrees of freedom. The "nite element analysis of vibrations of twisted blades based on beam theory was considered by Sisto and Chang [23]. Sabuncu and Thomas [24] studied the vibration characteristics of pre-twisted aerofoil cross-section blade packets under rotating conditions. An improved two-node Timoshenko beam "nite element was derived by Friedman and Kosmatka [25]. The vibration of Timoshenko beams with discontinuities in cross-section was investigated by Farghaly and Gadelab [26, 27]. Corn et al. [28] derived "nite element models through Guyan condensation method for the transverse vibration of short beams. Gupta and Rao [29] considered the "nite element analysis of tapered and twisted Timoshenko beams. In this work, the "nite element technique is applied to "nd the natural frequencies and mode shapes of beams in the bending}bending mode of vibration by taking into account the taper, the pre-twist and the rotation simultaneously. The coupling that exists between the #exural and torsional vibration is not considered. The taper and the angle of twist are assumed to vary linearly along the length of the beam. The element sti!ness and mass matrices are derived and the e!ects of o!set, rotation, pre-twist, depth and breadth taper ratios and rotary inertia and shear deformation on the natural frequencies are studied. Various special cases of beam vibration can be obtained from the general equations derived.
2. ELEMENT STIFFNESS AND MASS MATRICES
2.1. DISPLACEMENT MODEL Figure 1(a) shows a doubly tapered, twisted beam element of length l with the nodes as 1 and 2. The breadth, depth and the twist of the element are assumed to be linearly varying along its length. The breadth and depth at the two nodal points are shown as b , h and b , h respectively. The pre-twist at the two nodes is denoted by h and h . Figure 1(b) shows the nodal degrees of freedom of the element where bending de#ection, bending slope, shear de#ections and shear slope in the two planes are taken as the nodal degrees of freedom. Figure 1(c) shows the angle of twist h at any section z. The beam is assumed to rotate about the x}x-axis at a speed of X rad/s. The total de#ections of the element in the y and x directions at a distance z from node 1, w(z) and v(z), are taken as w(z)"w (z)#w (z), @ Q
v(z)"v (z)#v (z), @ Q
(1)
ROTATING TIMOSHENKO BEAMS
105
Figure 1. (a) An element of tapered and twisted beam, (b) degrees of freedom of an element, (c) angle of twist h, (d) rotation of tapered beam.
where w (z) and v (z) are the de#ections due to bending in the yz and xz planes respectively, @ @ and w (z) and v (z) are the de#ections due to shear in the corresponding planes. Q Q The displacement models for w (z), w (z), v (z) and v (z) are assumed to be polynomials of @ Q @ Q third degree. They are similar in nature except for the nodal constants. These expressions are given by u u u u w (z)" (2z!3lz#l)# (3lz!2z)! (z!2lz#lz)! (z!lz), @ l l l l u u u u w (z)" (2z!3lz!l)# (3lz!2z)! (z!2lz#lz)! (z!lz), Q l l l l
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S. S. RAO AND R. S. GUPTA
u u u v (z)" (2z!3lz!l)# (3lz!2z)! (z!2lz#lz) @ l l l (2) u ! (z!lz), l u u u v (z)" (2z!3lz!l)# (3lz!2z)! (z!2lz#lz) Q l l l u ! (z!lz), l where u , u , u and u represent the bending degrees of freedom and u , u , u and u are the shear degrees of freedom in the yz plane; u , u , u and u represent the bending degrees of freedom and u , u , u and u shear degrees of freedom in the xz plane. 2.2.
ELEMENT STIFFNESS MATRIX
The total strain energy ; of a beam of length l, due to bending and shear deformation including rotary inertia and rotation e!ects is given by
;"
J
EI *w *w *v EI *v VV @ #EI @ @# WW @ VW 2 *z *z *z 2 *z
kAG # 2
*w *v Q # Q *z *z
1 J *w *w @# Q dz dz# P(z) 2 *z *z
(3)
1 J *v *v J J @# Q dz! p (z) (w #w ) dz! p (z) (v #v ) dz. # P(z) U @ Q T @ Q 2 *z *z
where *>C
oAX
C>X >X mXm dm+ 2g
P(z)"
C
[(¸#e)!(e#z #z)] C (4)
oAX " g
1 1 1 e¸# ¸!ez ! z !(e#z )z! z , C 2 C C 2 2
oAX p (z)" (w #w ), U @ Q g
oAX p (z)" (v #v ), T @ Q g
(5, 6)
where e is the o!set and z is the distance of the "rst node of the element from the root of the C beam as shown in Figure 1(d), and P(z) is the axial force acting at section z.
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ROTATING TIMOSHENKO BEAMS
As the cross-section of the element changes with z and as the element is twisted, the cross-sectional area A, and the moments of inertia I , I and I will be functions of z: VV WW VW
z A(z)"b(z)h(z)" b #(b !b ) l
z h #(h !h ) l
(7)
1 " (c z#c lz#c l), l where c "(b !b ) (h !h ), c "b (h !h )#h (b !b ), c "b h , I (z)"I cos h#I sin h, VV VYVY WYWY I (z)"(I !I ) VW VYVY WYWY
(8)
I (z)"I cos h#I sin h, WW WYWY VYVY
sin 2h , 2
(9)
where x x and y y are the axes inclined at an angle h, the angle of twist, at any point in the element, to the original axes xx and yy as shown in Figure 1(c). The value of I "0 and the VYWY values of I and I can be computed as VYVY WYWY b(z)h(z) 1 " [a z#a lz#a lz#a lz#a l], I (z)" VYVY 12l 12
(10)
where a "(b !b ) (h !h ),
a "b (h !h )#3(b !b ) (h !h )h ,
a "3+b h (h !h )#(b !b ) (h !h )h ,
(11)
a "3b h (h !h )#(b !b )h , a "b h , I
WYWY
h(z)b(z) 1 (z)" " [d z#d lz#d lz#d lz#d l], 12 12l
(12)
where d "(h !h ) (b !b ), d "h (b !b )#3(h !h ) (b !b )b , d "3+h b (b !b )#(h !h ) (b !b )b ,
(13)
d "3h b (b !b )#(h !h )b , d "h b . By substituting the expressions of w , w , v , v , A, I , I and I from equations (2), (7) @ Q @ Q VV VW WW and (9) into equation (3), the strain energy ; can be expressed as ;" u2[K] u,
(14)
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S. S. RAO AND R. S. GUPTA
where u is the vector of nodal displacements u , u , 2 , u , and [K] is the elemental sti!ness matrix of order 16. Denoting the integrals
EIVV *z@ dz"[u u u u]2 [AK] [u u u u],
(15)
(16)
kAG *zQ dz"[u u u u]2 [CK] [u u u u],
(17)
(18)
P(z) *z@ dz"[u u u u]2 [EK] [u u u u]
(19)
J J
*w
*v @ dz"[u u u u ]2 [BK] [u u u u ], EI WW *z
J J
*w
*w @ *z
EI VW
J
*v @ dz"[u u u u ]2 [DK] [u u u u ], *z
*w
and J 2oAX (w) dz"[u u u u ]2 [FK] [u u u u ], @ g
(20)
the element sti!ness matrix can be expressed as [K]" [AK]#[EK]![FK]
[EK]![FK]
[DK]
[0]
[EK]![FK]
[CK]#[EK]![FK]
[0]
[0]
[DK]
[0]
[BK]#[EK]![FK]
[EK]![FK]
[0]
[0]
[EK]![FK]
[CK]#[EK]![FK]
,
(21) where [AK], [BK], [CK], [DK], +EK] and [FK] are symmetric matrices of order 4 and their elements are formulated in Appendix B. [0] is a null matrix of order 4.
2.3.
ELEMENT MASS MATRIX
The kinetic energy of the element ¹ including the e!ects of shear deformation and rotary inertia is given by
¹"
J oA *w *w oA *v *v oI *v @# Q # @# Q # WW @ 2g *t *t 2g *t *t 2g *z *t
oI *w @ # VW g *z *t
*v oI *w @ # VV @ dz. *z *t 2g *z *t
(22)
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ROTATING TIMOSHENKO BEAMS
By de"ning
g *t@ dz"[uR uR uR uR ]2 [AM] [uR uR uR uR ],
(23)
gVV *z *t@ dz"[uR uR uR uR ]2 [BM] [uR uR uR uR ],
(24)
gWW *z *t@ dz"[uR uR uR uR ]2 [CM] [uR uR uR uR ],
(25)
gVW *z *t@*z *t@ dz"[uR uR uR uR ]2 [DM] [uR uR uR uR ],
(26)
J oA *w
J oI
J oI
*w
*v
and J oI
*w
*v
where uR denotes the time derivative of the nodal displacement u , i"1, 2, 2 , 16, the J G kinetic energy of the element can be expressed as ¹" u [M] u ,
(27)
where [M] is the mass matrix given by [AM]#[BM] [AM] [M]" 16;16
[DM]
[0]
[AM]
[0]
[AM]
[AM]
[DM]
[AM] [AM]#[CM] [AM]
[0]
[0]
[AM]
(28)
[AM]
and [AM], [BM] [CM] and [DM] are symmetric matrices of order 4 whose elements are de"ned in Appendix A.
2.4.
BOUNDARY CONDITIONS
The following boundary conditions are to be applied depending on the type of end conditions. Free end
*w *v *v *w Q"0, Q"0, @"0, @"0. *z *z *z *z
Clamped end w "0, w "0, Q @ Hinged end w "0, Q
v "0, v "0, Q @
w "0, v "0, v "0. @ Q @
(29) *w *v @"0, @"0. *z *z
(30)
*v *w @"0, @"0. *z *z
(31)
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S. S. RAO AND R. S. GUPTA
TABLE 1 Natural frequencies (Hz) No of elements
First mode
Second mode
Third mode
Fourth mode
1 2 3 4 5 6 7 8
304.3 296)22 295)03 294)85 294)78 294)78 294)78 294)78
1191)66 1161)70 1155)97 1154)94 1154)67 1154)58 1154)54 1154)50
2327)07 1779)50 1746)55 1741)39 1739)99 1739)40 1739)25 1739)10
4552)23 4106)83 3747)04 3697)05 3689)82 3683)98 3683)91 3683)85
Data: length of beam"0)1524 m, breadth at root"0)0254 m, depth at root"0)0046 m, depth taper ratio"2)29, breadth taper ratio"2)56, twist angle"453, shear coe$cient"0)833, E"2)07;10 N/m, G"E/2)6, o!set"0, mass density"800 kg/m.
Figure 2. E!ect of depth taper ratio and shear deformation on frequency ratio of rotating beam: **, without shear deformation; } } } }, with shear deformation.
ROTATING TIMOSHENKO BEAMS
111
3. SPECIAL CASES
The various special cases of the beam vibration problem can be solved by applying one or more of the following four conditions: (a) For non-rotating beams: X"0 which results in [EK]"[FK]"[0].
(32)
(b) For uniform beams: by setting b "b and h "h , one obtains c "c "0 and c "b h , a "a "a "a "0 and a "b h , d "d "d "d "0 and
(33)
d "h b .
Figure 3. E!ect of breadth taper ratio and shear deformation on frequency ratio of rotating beam: **, with shear deformation; } } } }, without shear deformation.
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S. S. RAO AND R. S. GUPTA
(c) For neglecting the e!ect of shear deformation: w "v "0 so that equations (1) and (2) Q Q become w(z)"w (z) and v(z)"v (z). @ @
(34)
Due to this, the order of [K] and [M] matrices reduces from 16 to 8. (d) For beams without pre-twist: in this case, there will be no coupling between the moment of inertia terms and one obtains I "I , I "I , VV VYVY WW WYWY
I "0. VW
(35)
For vibration in the yz plane, v "v "0; for vibration in the xz plane, w "w "0. @ Q @ Q This condition further reduces the size of matrices [K] and [M] by half. Thus, for the general case (with shear deformation) the matrices will be of order 8 and, if coupled with condition (c) these will be of order 4. If the conditions (a), (b) and (d) are applied one gets the following expressions for [K] and [M] for non-rotating uniform beams without pre-twist but with a consideration of rotary inertia and shear deformation e!ects (for vibration in the y}z plane): 12
!12 !6l !6l 12
EI [K]" VV l
0
0
0
0
6l
6l
0
0
0
0
4l
2l
0
0
0
0
4l
0
0
0
0
36J !36J !3lJ !3lJ Symmetric
36J
3lJ
,
(36)
3lJ
4lJ !lJ 4lJ oAI [M]" 420g 156#36PM
54!36PM
22l!3lPM
156#36PM !13l#3lPM
13l!3lPM 156 22l#3lPM
4l#4lPM !3l!lPM 4l!4lPM
54
54
22l
13l
156 !13l
22l
22l !13l
4l
!3l
13l
22l
!3l
4l
156
54
22l
13l
156 !13l
22l
4l !3l 4l where J"kGb h l/(30EI ), and PM "14I /(b h l), VV VV
, (37)
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ROTATING TIMOSHENKO BEAMS
Equations (36) and (37) further reduce to the following well-known equations if the e!ects of shear deformation and rotary inertia are neglected: 12 EI [K]" VV l
!12 !6l !6l 12
6l
6l
Symmetric
4l
2l
,
(38)
4l 156 ob h [M]" 420g
54 !22l
13l
156 !13l
22l
Symmetric
4l !3l
.
(39)
4l
4. NUMERICAL RESULTS
The element sti!ness and mass matrices developed are used for the dynamic analysis of cantilever beams. By using the standard procedures of structural analysis, the eigenvalue
Figure 4. E!ect of rotation and twist on "rst and second natural frequencies.
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S. S. RAO AND R. S. GUPTA
problem can be stated as ([K ]!u[M ]) U"0, (40) where [K ] and [M ] denote the sti!ness and mass matrices of the structure, respectively, U indicates nodal displacement vector of the structure, and u is the natural frequency of vibration. A study of the convergence properties of the general element developed has been made and the results are given in Table 1. It is seen that the results converge well even with only four elements. The e!ects of shear deformation and depth taper ratio on the natural frequencies of a rotating twisted beam are shown in Figure 2 for a beam of length 0)254 m, o!set zero, depth at root 0)00865 m, breadth at root 0)0173 m, twist 453, rotation 100 r.p.s.
Figure 5. E!ect of rotation and twist on third and fourth natural frequencies: **, third mode; } } }, fourth mode.
ROTATING TIMOSHENKO BEAMS
115
and breadth taper ratio 3. The material properties of the beam are taken the same as those given in Table 1. The e!ect of shear deformation is found to reduce the frequencies at higher modes while at lower modes the results are nearly una!ected. There is an increase in the frequencies of vibration with an increase in the depth taper ratio in the "rst, second and fourth modes while a decrease has been observed in the case of third mode (vibration in a perpendicular plane). Figure 3 shows the variation of natural frequencies with breadth taper ratio. In this case the data are the same as in the case of Figure 2. In Figures 4 and 5, the variation of frequency ratio with rotation and pre-twist is studied. It can be seen that the frequency ratio changes slightly with the rotation but appreciably with the twist. At higher modes (in Figure 5) the e!ect of twist can be seen to be more pronounced. It is also observed that the frequency ratio increases with an increase in the twist in the case of "rst and third modes while it decreases with an increase of the twist in the case of second and fourth modes of vibration. In Figure 6, the e!ect of o!set is studied for a twisted blade having 603 twist with the other data the same as that of Figure 4. It is observed that an increase in o!set changes the frequency ratio more at higher values of rotation. The frequency ratio has been found to increase with an increase in the o!set.
Figure 6. E!ect of o!set and rotation on frequency ratio. O!set 1: e"0 m; o!set 2: e"0.0254 m; o!set 3: e"0.0508 m; o!set 4: e"0.0762 m.
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S. S. RAO AND R. S. GUPTA
5. CONCLUSION
The mass and sti!ness matrices of a thick rotating beam element with taper and twist are developed for the eigenvalue analysis of rotating, doubly tapered and twisted Timoshenko beams. The element has been found to give reasonably accurate results even with four "nite elements. The e!ects of breadth and depth taper ratios, twist angle, shear deformation, o!set and rotation on the natural frequencies of vibration of cantilever beams are found. The consideration of shear deformation is found to reduce the values of the higher natural frequencies of vibration of the beam. An increase in the breadth and depth taper ratios is found to increase the "rst two modes of vibration. The frequency ratio is found to change only slightly with rotation but appreciably with the twist of the beam.
REFERENCES 1. J. S. RAO 1965 Aeronautical Quarterly 16, 139. The fundament #exural vibration of cantilever beam of rectangular cross-section with uniform taper. 2. G. W. HOUSNER and W. O. KEIGHTLEY 1962 Proceedings of American Society of Civil Engineering 8, 95. Vibrations of linearly tapered beam. 3. J. S. RAO and W. CARNEGIE 1971 Bulletin of Mechanical Engineering Education 10, 239. Determination of the frequencies of lateral vibration of tapered cantilever beams by the use of Ritz}Galerkin process. 4. A. I. MARTIN 1956 Aeronautical Quarterly 7, 109. Some integrals relating to the vibration of a cantilever beams and approximations for the e!ect of taper on overtone frequencies. 5. H. H. MABIE and C. B. ROGERS 1972 Journal of Acoustical Society of America 51, (Part 2), 1771}1774. Transverse vibration of double-tapered cantilever beams. 6. A. MENDELSON and S. GENDLER 1949 NACA ¹N-2185. Analytical determination of coupled bending torsion vibrations of cantilever beams by means of station functions. 7. D. D. ROSARD 1953 Journal of Applied Mechanics American Society of Mechanical Engineers 20, 241. Natural frequencies of twisted cantilevers. 8. R. C. DI PRIMA and G. H. HANDELMAN 1954 Quarterly on Applied Mathematics 20, 241. Vibration of twisted beams. 9. W. CARNEGIE and J. THOMAS 1972 Journal of Engineering for Industry, ¹ransactions of American Society of Mechanical Engineers Series B, 94, 255}266. The coupled bending}bending vibration of pretwisted tapered blading. 10. J. S. RAO 1971 Journal of Aeronautical Society of India 23. Flexural vibration of pretwisted beams of rectangular cross section. 11. W. P. TARGOFF 1957 Proceedings of third Midwestern Conference on Solid Mechanics, 177. The bending vibrations of a twisted rotating beam. 12. R. T. YNTEMA 1955 NACA ¹N 3459. Simpli"ed procedures and charts for the rapid estimation of bending frequencies of rotating beams. 13. G. ISAKSON and G. J. EISLEY 1964 NASA No. NSG-27-59. Natural frequencies in coupled bending and torsion of twisted rotating and nonrotating blades. 14. S. BANERJEE and J. S. RAO American Society of Mechanical Engineers Paper No. 76-GT.43. Coupled bending-torsion vibrations of rotating blades. 15. W. CARNEGIE, C. STIRLING and J. FLEMING 1966 Paper No. 32. Applied Mechanics Convention, Cambridge, MA. Vibration characteristics of turbine blading under rotation-results of an initial investigation and details of a high-speed test installation. 16. R. M. KRUPKA and A. M. BAUMANIS 1969 Journal of Engineering for Industry American Society of Mechanical Engineers 91, 1017. Bending bending mode of a rotating tapered twisted turbomachine blade including rotary inertia and shear de#ection. 17. J. S. RAO and W. CARNEGIE 1973 International Journal of Mechanical Engineering Education 1, 37}47. Numerical procedure for the determination of the frequencies and mode shapes of lateral vibration of blades allowing for the e!ects of pretwist and rotation. 18. R. MCCALLEY 1963 Report No. DIG SA, 63}73, General Electric Company, Schenectady, New >ork. Rotary inertia correction for mass matrices. 19. K. K. KAPUR 1966 Journal of Acoustical Society of America 40, 1058}1063. Vibrations of a Timoshenko beam, using "nite element approach.
ROTATING TIMOSHENKO BEAMS
117
20. W. CARNEGIE, J. THOMAS and E. DOCUMAKI 1969 Aeronautical Quarterly 20, 321}332. An improved method of matrix displacement analysis in vibration problems. 21. R. NICKEL and G. SECOR 1972 International Journal of Numerical Methods in Engineering 5, 243}253. Convergence of consistently derived Timoshenko beam "nite elements. 22. J. THOMAS and B. A. H. ABBAS 1975 Journal of Sound and
APPENDIX A: EXPRESSIONS FOR STRAIN AND KINETIC ENERGIES A.1. EXPRESSION FOR STRAIN ENERGY (;)
A.1.1. Strain energy due to bending If the bending de#ections in yz and xz planes of a beam are w and v , respectively, the @ @ axial strain and stress induced due to w and v are given by e due to w "!y (*w /*z); @ @ VV @ @ e due to v "!x *v /*z and p due to w "!Ey *w /*z; p due to v " VV @ @ VV @ @ VV @ !Ex *v /*z. @ The strain energy stored in the beam due to bending is given by: ; due to bending 1 " 2
4 eVVpVV d<
1 " 2
4 (eVV due to w@#eVV due to v@) (pVV due to w@#pVV due to v@) d<
*w *v E J @#x @ dA " dz y 2 *z *z
*z@ y dA# *z@ x dA#2 *z@ *z@ xy dA
E J " dz 2
*w
*v
*w *v
2VV *z@ #EIVW *z@ *z@# 2WW *z@ dz,
"
J EI
*w
*w *v
EI
*v
where < is the volume, l is the length and A is the cross-sectional area of the beam.
(A.1)
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S. S. RAO AND R. S. GUPTA
A.1.2. Strain energy due to shearing Let F and F be the shear forces that produce the shear de#ections dv and dw in an V W Q Q element of length dz respectively. Then the strain energy of the beam due to shearing is given by
1 J dv dw Q#F Q dz. ; " F BSC RM QFC?PGLE 2 V dz W dz By substituting AGk dv /dz and AGk dw /dz for F and F , respectively, one obtains Q Q V W J kAG 2
; " BSC RM QFC?PGLE
dw dv Q # Q dz. dz dz
(A.2)
A.1.3. Strain energy due to rotation The rotation of a beam induces an axial force P in the beam due to centrifugal action. If the beam is bending in the yz plane (Figure A1), the change in the horizontal projection of an element of length ds is given by
ds!dz" (dz)#
*w 1 *w dz !dz+ dz. *z 2 *z
Since the axial force P acts against the changes in the horizontal projection, the work done by P is given by
dw 1 J P(z) dz. ; "! BSC RM . ?LB U 2 dz
(A.3)
The work done by the transverse distributed force p (z) can be written as U J
pU (z) w dz.
; " BSC RM NU
Figure A.1. An element of the beam in equilibrium.
(A.4)
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ROTATING TIMOSHENKO BEAMS
The expressions corresponding to the bending of the beam in the xz plane can be obtained similarly as
1 J dv ; " P(z) dz, BSC RM . ?LB T 2 dz
J
pT (z) v dz.
" ; BSC RM NT
(A.5, 6)
The total strain energy of the beam can be obtained as given in equation (3) by combining equations (A.1)}(A.6).
A.2.
EXPRESSION FOR KINETIC ENERGY (¹)
Consider a small element of area dA and length dz at a point in the cross-section having co-ordinates (x, y) with respect to x- and y-axes. The kinetic energy of this element is given by o [(wR #vR )#(y Q #x Q )] dz dA, V W 2g where and denote the bending slopes, *v /*z and *w /*z, respectively, and a dot over V W @ @ a symbol represents derivative with respect to time. Integrating this equation over the beam cross-section, the kinetic energy of an element of length dz can be obtained as o d¹" [A(wR #vR )#(I Q #2I Q Q #I Q )] dz. VV W VW V W WW V 2g The kinetic energy of the entire beam (¹ ) can be expressed as
¹"
J oA o (wR #vR )# (I Q #2I Q Q #I Q )] dz, VW V W WW V 2g VV W 2g
(A.7)
which can be seen to be the same as in equation (22).
APPENDIX B: EXPRESSIONS FOR [AK], +BK], 2 , [DM]
The following notation is used for convenience: J
zG\ dz,
;" G
¸ "lG\, i"1, 2, 2 , n, G
zG\ cos (h!h) l #h dz,
<" G
J J
S" G
z
(B.1, 2)
i"1, 2, 2 , n,
(B.3)
z zG\ sin 2 (h !h ) #h dz, i"1, 2, 2 , n, l
(B.4)
where h and h denote the values of pre-twist at nodes 1 and 2, respectively, of the element. As the nature of w , w , v and v is the same except for their positions in the sti!ness and @ Q @ Q mass matrices, one can use wN to denote any one of the quantities w , w , v or v and in @ Q @ Q
120
S. S. RAO AND R. S. GUPTA
TABLE B1
j
1 1 1 1 2 2 2 3 3 4
1 2 3 4 2 3 4 3 4 4
H G H
Q G H I
G H I
k"1
k"2
0 144)0 0 !144)0 1 !72)0 1 !72)0 0 144)0 1 72)0 1 72)0 2 36)0 2 36)0 2 36)0
!144)0 144)0 84)0 60)0 !144)0 !84)0 !60)0 !48)0 !36)0 !24)0
k"3 36)0 !36)0 !24)0 !12)0 36)0 24)0 12)0 16)0 8)0 4)0
k"1 36)0 !36)0 !18)0 !18)0 36)0 18)0 18)0 9)0 9)0 9)0
k"2
k"3
!72)0 72)0 42)0 30)0 !72)0 !42)0 !30)0 !24)0 !18)0 !12)0
36)0 !36)0 !30)0 !12)0 36)0 30)0 12)0 22)0 11)0 4)0
k"4 0)0 0)0 6)0 0)0 0)0 !6)0 0)0 !8)0 !2)0 0)0
k"5 0)0 0)0 0)0 0)0 0)0 0)0 0)0 1)0 0)0 0)0
a similar manner the set (uN , uN , uN , uN ) can be used to represent any one of the sets (u , u , u , u ), (u , u , u , u ), (u , u , u , u ) or (u , u , u , u ). Thus, uN uN uN uN wN (z)" (2z!3lz#l)! (z!2lz#lz)# (3lz!2z)! (z!lz), (B.5) l l l l dwN uN uN uN uN " (6z!6lz)! (3z!4lz#l)# (6lz!6z)! (3z!2lz), dz l l l l dwN uN uN uN uN " (12z!6l)! (6z!4l)# (6l!12z)! (6z!2l). dz l l l l
(B.6)
(B.7)
By letting P (i"1,2, 4; j"1,2, 4; k"1,2, 7) denote the coe$cient of zI\ l\I for G H I the uN uN term in the expression of wN , Q (i"1,2, 4; j"1,2, 4; k"1,2, 5), the G H G H I coe$cient of zI\ l\I for the uN uN term in the expression of (dwN /dz), R (i"1,2, 4; G H G H I j"1,2, 4; k"1,2, 3), the coe$cient of zI\ l\I for the uN uN term in the expression of G H (dwN /dz), H (i"1,2, 4; j"1,2, 4), the index coe$cient of l to account for the G H di!erence in the index of l due to multiplication of rotational degrees of freedom uN and uN and the displacement degrees of freedom uN and uN , the values of P , Q ,R and G H I G H I G H I H can be obtained as shown in Tables B1 and B2. G H B.1.
EVALUATION OF [BK]
As the procedure for the derivation of [AK], [BK],2, [DM] is the same, the expression for [BK] is derived here as an illustration:
EIWW *z@ dz
[u u u u ]2 [BK] [u u u u ]"
"
J
*wN EI dz, WW *z
J
*v
(B.8)
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ROTATING TIMOSHENKO BEAMS
TABLE B2
j
k"1
1 1 1 1 2 2 2 3 3 4
1 2 3 4 2 3 4 3 4 4
4)0 !4)0 !2)0 !2)0 4)0 2)0 2)0 1)0 1)0 1)0
k"2 !12)0 12)0 7)0 5)0 !12)0 !7)0 !5)0 !4)0 !3)0 !2)0
k"3
k"4
9)0 !9)0 !8)0 !3)0 9)0 8)0 3)0 6)0 3)0 1)0
k"5
4)0 !2)0 2)0 !1)0 0)0 !3)0 0)0 !4)0 !1)0 0)0
k"6 0)0 0)0 !1)0 0)0 0)0 0)0 0)0 0)0 0)0 0)0
!6)0 3)0 2)0 1)0 0)0 0)0 0)0 1)0 0)0 0)0
k"7 1)0 0)0 0)0 0)0 0)0 0)0 0)0 0)0 0)0 0)0
where
wN "v and @
uN u uN u " . uN u uN u
Substituting the values of I and wN into equation (B.8), WW
*wN J z EI dz" E I #(I !I ) cos (h !h ) #h WW *z VYVY WYWY VYVY l
J
uN uN uN uN (12z!6l)! (6z!4l)# (6l!12z)! (6z!2l) dz l l l l
with BK "coe$cient of uN uN "coe$cient of u u
(az#alz#alz#alz#al)
E J " l&G H 12l
#+(d z#d lz#d lz#d lz#d l)
z !(a z#a lz#a lz#a lz#a l), cos (h !h ) #h l
;+R z#R lz#R l, dz
(B.9)
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S. S. RAO AND R. S. GUPTA
E " ¸ a [R ; #¸ R ; #¸ R < ] & G \G G> \G G> \G > 12l G (d !a ) [¸ R < #¸ R < #¸ R < ] G G G \G G> \G G> \G
(B.10)
E +a ¸ ; R ,#(d !a ) +¸ < R ,. " ¸ G G>H\ \G\H H G G G>H\ \G\H H 12l & > G H This relation can be generalized as E BK " ¸ +a ¸ ; R , ' ( 12l & (> G G>H\ \G\H ' ( H G H #(d !a ) +¸ < R ,, I"1,2, 4, J"I,2, 4, G G G>H\ \G\H ' ( H
(B.11)
E " [+a ¸ ; R , G G>H>&' ( \G\H ' ( H 12l G H < R ,], I"1,2, 4, #(d !a ) +¸ G G G>H>&' ( \G\H ' ( H
J"1,2, 4.
Similarly, E AK " [+d ¸ ; R , ' ( 12l G G>H>&' ( \G\H ' ( H G H
(B.12)
< R ,], I"1,2, 4, J"1,2, 4, #(a !d ) +¸ G G G>H>&' ( \G\H ' ( H kG CK " [c ¸ ; Q ], ' ( l G G>H>&' ( \G\H ' ( H G H
I"1,2, 4,
J"1,2, 4,
E DK " [(a !d )¸ S R ], I"1,2, 4, ' ( 12l G G G>H>&' ( \G\H ' ( H G H o [c ¸ ; P ], I"1,2, 4, AM " G G>H>&' ( \G\H ' ( H ' ( gl G H
J"1,2, 4,
J"1,2, 4,
o BM " [+d ¸ ; Q , ' ( 12gl G G>H>&' ( \G\H ' ( H G H #(a !d ) +¸ < Q ,], G G G>H>&' ( \G\H ' ( H
(B.14) (B.15)
(B.16)
I"1,2, 4, J"1,2, 4,
o CM " [+a ¸ ; Q , ' ( 12gl G G>H>&' ( \G\H ' ( H G H #(d !a ) +¸ < Q ,], G G G>H>&' ( \G\H ' ( H
(B.13)
I"1,2, 4, J"1,2, 4,
(B.17)
123
ROTATING TIMOSHENKO BEAMS
o DM " [(a !d ) ¸ S Q ], ' ( 12gl G G G>H>&' ( \G\H ' ( H G H
(B.18)
I"1,2, 4, J"1,2, 4.
B.2.
EVALUATION OF [EK] AND [FK]
[u u u u ]2 [EK] [u u u u ]" J oAX g
"
P(z)
*w @ dz *z
1 1 e¸# ¸!ez !z !(e#z ) z! z C C C 2 2
1 J oAX e¸# ¸!ez !z " C C 2 g
Thus,
J
*w @ dz *z (B.19)
*w @ dz *z
J oAX *w J 1 oAX *w @ dz! (e#z ) z z ! C g *z 2 g *z@ dz.
1 oX e¸# ¸!ez !z EK " C C ' ( 2 gl
[C ¸ ; Q ] G G>H>&' ( \G\H ' ( H G H
oX ! (e#z ) [C ¸ ; Q ] C G G>H>&' ( \G\H ' ( H gl G H oX ! [C ¸ ; Q ], I"1,2, 4, G G>H>&' ( \G\H ' ( H 2gl G H Similarly,
(B.20) J"1,2, 4.
2oX FK " ; P ], I"1,2, 4, J"I,2, 4. (B.21) [C ¸ ' ( G G>H>&' ( \G\H ' ( H gl G H
APPENDIX C: NOMENCLATURE A b e E g G h I ,I ,I VV WW VW [K] l ¸ [M] t u
area of cross-section breadth of beam o!set Young's modulus acceleration due to gravity shear modulus depth of beam moment of inertia of beam cross-section about xx-, yy- and xy-axis element sti!ness matrix length of an element length of total beam element mass matrix time parameter nodal degrees of freedom
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S. S. RAO AND R. S. GUPTA
a b h o k X
strain energy displacement in xz plane displacement in yz plane co-ordinate axes co-ordinate axis and length parameter distance of the "rst node of the element from the root of the beam ratio of modal frequency to frequency of fundamental mode of uniform beam with the same root cross-section and without shear deformation e!ects depth taper ratio"h /h breadth taper ratio"b /b angle of twist weight density shear coe$cient rotational speed of the beam (rad/s)
Subscripts b s
bending shear
; v w x, y z z C frequency ratio
FINITE ELEMENT VIBRATION ANALYSIS OF ROTATING TIMOSHENKO BEAMS S. S. RAO Department of Mechanical Engineering, ;niversity of Miami, Coral Gables, F¸ 33124-0624, ;.S.A. AND
R. S. GUPTA Department of Mechanical Engineering, Punjab Engineering College, Chandigarh 11, India (Received 12 January 2000, and in ,nal form 13 September 2000) The sti!ness and mass matrices of a rotating twisted and tapered beam element are derived. The angle of twist, breadth and depth are assumed to vary linearly along the length of beam. The e!ects of shear deformation and rotary inertia are also considered in deriving the elemental matrices. The "rst four natural frequencies and mode shapes in bending}bending mode are calculated for cantilever beams. The e!ects of twist, o!set, speed of rotation and variation of depth and breadth taper ratios are studied. 2001 Academic Press 1. INTRODUCTION
Generally, turbomachine blades are idealized as tapered cantilever beams. In order to re"ne the analysis the e!ects of pre-twist and rotation are also included. As the blades are short in some of the designs and may vibrate in higher-frequency ranges, the e!ects of shear deformation and rotary inertia may be of considerable magnitude. Various investigators in the "eld of turbine-blade vibrations have solved the di!erential equations of motion, by taking into account one or more of the above-mentioned e!ects. The analysis of tapered beams has been made by many investigators using di!erent techniques. Rao [1] used the Galerkin method, Housner and Keightley [2] applied the Myklestad procedure, Rao and Carnegie [3] used the "nite di!erences approach, Martin [4] adopted a perturbation technique and Mabie and Rogers [5] solved the di!erential equations using Bessel functions to "nd the natural frequencies of vibration of tapered cantilever beams. In analysing the pre-twisted beams, various approaches have been used by various investigators. Mendelson and Gendler [6] used the station functions, Rosard [7] applied the Myklestad method, Di Prima and Handleman [8] adopted the Rayleigh}Ritz procedure, Carnegie and Thomas [9] used the "nite di!erence techniques and Rao [10] considered the Galerkin method for the analysis of pre-twisted beams. The bending vibration of a twisted rotating beam was considered by Targo! [11]. The rotation e!ect in beam analysis has also been considered using di!erent formulation and solution procedures. Targo! applied the matrix method, Yntema [12] applied the Rayleigh}Ritz procedure, Isakson and Eisley [13] used the Rayleigh}Southwell procedure, Banerjee and Rao [14] applied the Galerkin procedure, Carnegie et al. [15] applied the "nite di!erence scheme, Krupka and Baumanis [16] used the Myklestad method and Rao 0022-460X/01/160103#22 $35.00/0
2001 Academic Press
104
S. S. RAO AND R. S. GUPTA
and Carnegie [17] used the extended Holzer method to "nd the vibration characteristics of rotating cantilever beams. The "nite element technique has also been applied by many investigators, mostly for the vibration analysis of beams of uniform cross-section. All these investigations di!er from one another in the nodal degrees of freedom taken for deriving the elemental sti!ness and mass matrices. McCalley [18] derived the consistent mass and sti!ness matrices by selecting the total de#ection and total slope as nodal co-ordinates. Kapur [19] took bending de#ection, shear de#ection, bending slope and shear slope as nodal degrees of freedom and derived the elemental matrices of beams with linearly varying inertia. Carnegie et al. [20] analyzed uniform beams by taking few internal nodes in it. Nickel and Sector [21] used total de#ection, total slope and bending slope of the two nodes and the bending slope at the mid-point of the beam as the degrees of freedom to derive the elemental sti!ness and the mass matrices of order seven. Thomas and Abbas [22] analyzed uniform Timoshenko beams by taking total de#ection, total slope, bending slope and the derivative of the bending slope as nodal degrees of freedom. The "nite element analysis of vibrations of twisted blades based on beam theory was considered by Sisto and Chang [23]. Sabuncu and Thomas [24] studied the vibration characteristics of pre-twisted aerofoil cross-section blade packets under rotating conditions. An improved two-node Timoshenko beam "nite element was derived by Friedman and Kosmatka [25]. The vibration of Timoshenko beams with discontinuities in cross-section was investigated by Farghaly and Gadelab [26, 27]. Corn et al. [28] derived "nite element models through Guyan condensation method for the transverse vibration of short beams. Gupta and Rao [29] considered the "nite element analysis of tapered and twisted Timoshenko beams. In this work, the "nite element technique is applied to "nd the natural frequencies and mode shapes of beams in the bending}bending mode of vibration by taking into account the taper, the pre-twist and the rotation simultaneously. The coupling that exists between the #exural and torsional vibration is not considered. The taper and the angle of twist are assumed to vary linearly along the length of the beam. The element sti!ness and mass matrices are derived and the e!ects of o!set, rotation, pre-twist, depth and breadth taper ratios and rotary inertia and shear deformation on the natural frequencies are studied. Various special cases of beam vibration can be obtained from the general equations derived.
2. ELEMENT STIFFNESS AND MASS MATRICES
2.1. DISPLACEMENT MODEL Figure 1(a) shows a doubly tapered, twisted beam element of length l with the nodes as 1 and 2. The breadth, depth and the twist of the element are assumed to be linearly varying along its length. The breadth and depth at the two nodal points are shown as b , h and b , h respectively. The pre-twist at the two nodes is denoted by h and h . Figure 1(b) shows the nodal degrees of freedom of the element where bending de#ection, bending slope, shear de#ections and shear slope in the two planes are taken as the nodal degrees of freedom. Figure 1(c) shows the angle of twist h at any section z. The beam is assumed to rotate about the x}x-axis at a speed of X rad/s. The total de#ections of the element in the y and x directions at a distance z from node 1, w(z) and v(z), are taken as w(z)"w (z)#w (z), @ Q
v(z)"v (z)#v (z), @ Q
(1)
ROTATING TIMOSHENKO BEAMS
105
Figure 1. (a) An element of tapered and twisted beam, (b) degrees of freedom of an element, (c) angle of twist h, (d) rotation of tapered beam.
where w (z) and v (z) are the de#ections due to bending in the yz and xz planes respectively, @ @ and w (z) and v (z) are the de#ections due to shear in the corresponding planes. Q Q The displacement models for w (z), w (z), v (z) and v (z) are assumed to be polynomials of @ Q @ Q third degree. They are similar in nature except for the nodal constants. These expressions are given by u u u u w (z)" (2z!3lz#l)# (3lz!2z)! (z!2lz#lz)! (z!lz), @ l l l l u u u u w (z)" (2z!3lz!l)# (3lz!2z)! (z!2lz#lz)! (z!lz), Q l l l l
106
S. S. RAO AND R. S. GUPTA
u u u v (z)" (2z!3lz!l)# (3lz!2z)! (z!2lz#lz) @ l l l (2) u ! (z!lz), l u u u v (z)" (2z!3lz!l)# (3lz!2z)! (z!2lz#lz) Q l l l u ! (z!lz), l where u , u , u and u represent the bending degrees of freedom and u , u , u and u are the shear degrees of freedom in the yz plane; u , u , u and u represent the bending degrees of freedom and u , u , u and u shear degrees of freedom in the xz plane. 2.2.
ELEMENT STIFFNESS MATRIX
The total strain energy ; of a beam of length l, due to bending and shear deformation including rotary inertia and rotation e!ects is given by
;"
J
EI *w *w *v EI *v VV @ #EI @ @# WW @ VW 2 *z *z *z 2 *z
kAG # 2
*w *v Q # Q *z *z
1 J *w *w @# Q dz dz# P(z) 2 *z *z
(3)
1 J *v *v J J @# Q dz! p (z) (w #w ) dz! p (z) (v #v ) dz. # P(z) U @ Q T @ Q 2 *z *z
where *>C
oAX
C>X >X mXm dm+ 2g
P(z)"
C
[(¸#e)!(e#z #z)] C (4)
oAX " g
1 1 1 e¸# ¸!ez ! z !(e#z )z! z , C 2 C C 2 2
oAX p (z)" (w #w ), U @ Q g
oAX p (z)" (v #v ), T @ Q g
(5, 6)
where e is the o!set and z is the distance of the "rst node of the element from the root of the C beam as shown in Figure 1(d), and P(z) is the axial force acting at section z.
107
ROTATING TIMOSHENKO BEAMS
As the cross-section of the element changes with z and as the element is twisted, the cross-sectional area A, and the moments of inertia I , I and I will be functions of z: VV WW VW
z A(z)"b(z)h(z)" b #(b !b ) l
z h #(h !h ) l
(7)
1 " (c z#c lz#c l), l where c "(b !b ) (h !h ), c "b (h !h )#h (b !b ), c "b h , I (z)"I cos h#I sin h, VV VYVY WYWY I (z)"(I !I ) VW VYVY WYWY
(8)
I (z)"I cos h#I sin h, WW WYWY VYVY
sin 2h , 2
(9)
where x x and y y are the axes inclined at an angle h, the angle of twist, at any point in the element, to the original axes xx and yy as shown in Figure 1(c). The value of I "0 and the VYWY values of I and I can be computed as VYVY WYWY b(z)h(z) 1 " [a z#a lz#a lz#a lz#a l], I (z)" VYVY 12l 12
(10)
where a "(b !b ) (h !h ),
a "b (h !h )#3(b !b ) (h !h )h ,
a "3+b h (h !h )#(b !b ) (h !h )h ,
(11)
a "3b h (h !h )#(b !b )h , a "b h , I
WYWY
h(z)b(z) 1 (z)" " [d z#d lz#d lz#d lz#d l], 12 12l
(12)
where d "(h !h ) (b !b ), d "h (b !b )#3(h !h ) (b !b )b , d "3+h b (b !b )#(h !h ) (b !b )b ,
(13)
d "3h b (b !b )#(h !h )b , d "h b . By substituting the expressions of w , w , v , v , A, I , I and I from equations (2), (7) @ Q @ Q VV VW WW and (9) into equation (3), the strain energy ; can be expressed as ;" u2[K] u,
(14)
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S. S. RAO AND R. S. GUPTA
where u is the vector of nodal displacements u , u , 2 , u , and [K] is the elemental sti!ness matrix of order 16. Denoting the integrals
EIVV *z@ dz"[u u u u]2 [AK] [u u u u],
(15)
(16)
kAG *zQ dz"[u u u u]2 [CK] [u u u u],
(17)
(18)
P(z) *z@ dz"[u u u u]2 [EK] [u u u u]
(19)
J J
*w
*v @ dz"[u u u u ]2 [BK] [u u u u ], EI WW *z
J J
*w
*w @ *z
EI VW
J
*v @ dz"[u u u u ]2 [DK] [u u u u ], *z
*w
and J 2oAX (w) dz"[u u u u ]2 [FK] [u u u u ], @ g
(20)
the element sti!ness matrix can be expressed as [K]" [AK]#[EK]![FK]
[EK]![FK]
[DK]
[0]
[EK]![FK]
[CK]#[EK]![FK]
[0]
[0]
[DK]
[0]
[BK]#[EK]![FK]
[EK]![FK]
[0]
[0]
[EK]![FK]
[CK]#[EK]![FK]
,
(21) where [AK], [BK], [CK], [DK], +EK] and [FK] are symmetric matrices of order 4 and their elements are formulated in Appendix B. [0] is a null matrix of order 4.
2.3.
ELEMENT MASS MATRIX
The kinetic energy of the element ¹ including the e!ects of shear deformation and rotary inertia is given by
¹"
J oA *w *w oA *v *v oI *v @# Q # @# Q # WW @ 2g *t *t 2g *t *t 2g *z *t
oI *w @ # VW g *z *t
*v oI *w @ # VV @ dz. *z *t 2g *z *t
(22)
109
ROTATING TIMOSHENKO BEAMS
By de"ning
g *t@ dz"[uR uR uR uR ]2 [AM] [uR uR uR uR ],
(23)
gVV *z *t@ dz"[uR uR uR uR ]2 [BM] [uR uR uR uR ],
(24)
gWW *z *t@ dz"[uR uR uR uR ]2 [CM] [uR uR uR uR ],
(25)
gVW *z *t@*z *t@ dz"[uR uR uR uR ]2 [DM] [uR uR uR uR ],
(26)
J oA *w
J oI
J oI
*w
*v
and J oI
*w
*v
where uR denotes the time derivative of the nodal displacement u , i"1, 2, 2 , 16, the J G kinetic energy of the element can be expressed as ¹" u [M] u ,
(27)
where [M] is the mass matrix given by [AM]#[BM] [AM] [M]" 16;16
[DM]
[0]
[AM]
[0]
[AM]
[AM]
[DM]
[AM] [AM]#[CM] [AM]
[0]
[0]
[AM]
(28)
[AM]
and [AM], [BM] [CM] and [DM] are symmetric matrices of order 4 whose elements are de"ned in Appendix A.
2.4.
BOUNDARY CONDITIONS
The following boundary conditions are to be applied depending on the type of end conditions. Free end
*w *v *v *w Q"0, Q"0, @"0, @"0. *z *z *z *z
Clamped end w "0, w "0, Q @ Hinged end w "0, Q
v "0, v "0, Q @
w "0, v "0, v "0. @ Q @
(29) *w *v @"0, @"0. *z *z
(30)
*v *w @"0, @"0. *z *z
(31)
110
S. S. RAO AND R. S. GUPTA
TABLE 1 Natural frequencies (Hz) No of elements
First mode
Second mode
Third mode
Fourth mode
1 2 3 4 5 6 7 8
304.3 296)22 295)03 294)85 294)78 294)78 294)78 294)78
1191)66 1161)70 1155)97 1154)94 1154)67 1154)58 1154)54 1154)50
2327)07 1779)50 1746)55 1741)39 1739)99 1739)40 1739)25 1739)10
4552)23 4106)83 3747)04 3697)05 3689)82 3683)98 3683)91 3683)85
Data: length of beam"0)1524 m, breadth at root"0)0254 m, depth at root"0)0046 m, depth taper ratio"2)29, breadth taper ratio"2)56, twist angle"453, shear coe$cient"0)833, E"2)07;10 N/m, G"E/2)6, o!set"0, mass density"800 kg/m.
Figure 2. E!ect of depth taper ratio and shear deformation on frequency ratio of rotating beam: **, without shear deformation; } } } }, with shear deformation.
ROTATING TIMOSHENKO BEAMS
111
3. SPECIAL CASES
The various special cases of the beam vibration problem can be solved by applying one or more of the following four conditions: (a) For non-rotating beams: X"0 which results in [EK]"[FK]"[0].
(32)
(b) For uniform beams: by setting b "b and h "h , one obtains c "c "0 and c "b h , a "a "a "a "0 and a "b h , d "d "d "d "0 and
(33)
d "h b .
Figure 3. E!ect of breadth taper ratio and shear deformation on frequency ratio of rotating beam: **, with shear deformation; } } } }, without shear deformation.
112
S. S. RAO AND R. S. GUPTA
(c) For neglecting the e!ect of shear deformation: w "v "0 so that equations (1) and (2) Q Q become w(z)"w (z) and v(z)"v (z). @ @
(34)
Due to this, the order of [K] and [M] matrices reduces from 16 to 8. (d) For beams without pre-twist: in this case, there will be no coupling between the moment of inertia terms and one obtains I "I , I "I , VV VYVY WW WYWY
I "0. VW
(35)
For vibration in the yz plane, v "v "0; for vibration in the xz plane, w "w "0. @ Q @ Q This condition further reduces the size of matrices [K] and [M] by half. Thus, for the general case (with shear deformation) the matrices will be of order 8 and, if coupled with condition (c) these will be of order 4. If the conditions (a), (b) and (d) are applied one gets the following expressions for [K] and [M] for non-rotating uniform beams without pre-twist but with a consideration of rotary inertia and shear deformation e!ects (for vibration in the y}z plane): 12
!12 !6l !6l 12
EI [K]" VV l
0
0
0
0
6l
6l
0
0
0
0
4l
2l
0
0
0
0
4l
0
0
0
0
36J !36J !3lJ !3lJ Symmetric
36J
3lJ
,
(36)
3lJ
4lJ !lJ 4lJ oAI [M]" 420g 156#36PM
54!36PM
22l!3lPM
156#36PM !13l#3lPM
13l!3lPM 156 22l#3lPM
4l#4lPM !3l!lPM 4l!4lPM
54
54
22l
13l
156 !13l
22l
22l !13l
4l
!3l
13l
22l
!3l
4l
156
54
22l
13l
156 !13l
22l
4l !3l 4l where J"kGb h l/(30EI ), and PM "14I /(b h l), VV VV
, (37)
113
ROTATING TIMOSHENKO BEAMS
Equations (36) and (37) further reduce to the following well-known equations if the e!ects of shear deformation and rotary inertia are neglected: 12 EI [K]" VV l
!12 !6l !6l 12
6l
6l
Symmetric
4l
2l
,
(38)
4l 156 ob h [M]" 420g
54 !22l
13l
156 !13l
22l
Symmetric
4l !3l
.
(39)
4l
4. NUMERICAL RESULTS
The element sti!ness and mass matrices developed are used for the dynamic analysis of cantilever beams. By using the standard procedures of structural analysis, the eigenvalue
Figure 4. E!ect of rotation and twist on "rst and second natural frequencies.
114
S. S. RAO AND R. S. GUPTA
problem can be stated as ([K ]!u[M ]) U"0, (40) where [K ] and [M ] denote the sti!ness and mass matrices of the structure, respectively, U indicates nodal displacement vector of the structure, and u is the natural frequency of vibration. A study of the convergence properties of the general element developed has been made and the results are given in Table 1. It is seen that the results converge well even with only four elements. The e!ects of shear deformation and depth taper ratio on the natural frequencies of a rotating twisted beam are shown in Figure 2 for a beam of length 0)254 m, o!set zero, depth at root 0)00865 m, breadth at root 0)0173 m, twist 453, rotation 100 r.p.s.
Figure 5. E!ect of rotation and twist on third and fourth natural frequencies: **, third mode; } } }, fourth mode.
ROTATING TIMOSHENKO BEAMS
115
and breadth taper ratio 3. The material properties of the beam are taken the same as those given in Table 1. The e!ect of shear deformation is found to reduce the frequencies at higher modes while at lower modes the results are nearly una!ected. There is an increase in the frequencies of vibration with an increase in the depth taper ratio in the "rst, second and fourth modes while a decrease has been observed in the case of third mode (vibration in a perpendicular plane). Figure 3 shows the variation of natural frequencies with breadth taper ratio. In this case the data are the same as in the case of Figure 2. In Figures 4 and 5, the variation of frequency ratio with rotation and pre-twist is studied. It can be seen that the frequency ratio changes slightly with the rotation but appreciably with the twist. At higher modes (in Figure 5) the e!ect of twist can be seen to be more pronounced. It is also observed that the frequency ratio increases with an increase in the twist in the case of "rst and third modes while it decreases with an increase of the twist in the case of second and fourth modes of vibration. In Figure 6, the e!ect of o!set is studied for a twisted blade having 603 twist with the other data the same as that of Figure 4. It is observed that an increase in o!set changes the frequency ratio more at higher values of rotation. The frequency ratio has been found to increase with an increase in the o!set.
Figure 6. E!ect of o!set and rotation on frequency ratio. O!set 1: e"0 m; o!set 2: e"0.0254 m; o!set 3: e"0.0508 m; o!set 4: e"0.0762 m.
116
S. S. RAO AND R. S. GUPTA
5. CONCLUSION
The mass and sti!ness matrices of a thick rotating beam element with taper and twist are developed for the eigenvalue analysis of rotating, doubly tapered and twisted Timoshenko beams. The element has been found to give reasonably accurate results even with four "nite elements. The e!ects of breadth and depth taper ratios, twist angle, shear deformation, o!set and rotation on the natural frequencies of vibration of cantilever beams are found. The consideration of shear deformation is found to reduce the values of the higher natural frequencies of vibration of the beam. An increase in the breadth and depth taper ratios is found to increase the "rst two modes of vibration. The frequency ratio is found to change only slightly with rotation but appreciably with the twist of the beam.
REFERENCES 1. J. S. RAO 1965 Aeronautical Quarterly 16, 139. The fundament #exural vibration of cantilever beam of rectangular cross-section with uniform taper. 2. G. W. HOUSNER and W. O. KEIGHTLEY 1962 Proceedings of American Society of Civil Engineering 8, 95. Vibrations of linearly tapered beam. 3. J. S. RAO and W. CARNEGIE 1971 Bulletin of Mechanical Engineering Education 10, 239. Determination of the frequencies of lateral vibration of tapered cantilever beams by the use of Ritz}Galerkin process. 4. A. I. MARTIN 1956 Aeronautical Quarterly 7, 109. Some integrals relating to the vibration of a cantilever beams and approximations for the e!ect of taper on overtone frequencies. 5. H. H. MABIE and C. B. ROGERS 1972 Journal of Acoustical Society of America 51, (Part 2), 1771}1774. Transverse vibration of double-tapered cantilever beams. 6. A. MENDELSON and S. GENDLER 1949 NACA ¹N-2185. Analytical determination of coupled bending torsion vibrations of cantilever beams by means of station functions. 7. D. D. ROSARD 1953 Journal of Applied Mechanics American Society of Mechanical Engineers 20, 241. Natural frequencies of twisted cantilevers. 8. R. C. DI PRIMA and G. H. HANDELMAN 1954 Quarterly on Applied Mathematics 20, 241. Vibration of twisted beams. 9. W. CARNEGIE and J. THOMAS 1972 Journal of Engineering for Industry, ¹ransactions of American Society of Mechanical Engineers Series B, 94, 255}266. The coupled bending}bending vibration of pretwisted tapered blading. 10. J. S. RAO 1971 Journal of Aeronautical Society of India 23. Flexural vibration of pretwisted beams of rectangular cross section. 11. W. P. TARGOFF 1957 Proceedings of third Midwestern Conference on Solid Mechanics, 177. The bending vibrations of a twisted rotating beam. 12. R. T. YNTEMA 1955 NACA ¹N 3459. Simpli"ed procedures and charts for the rapid estimation of bending frequencies of rotating beams. 13. G. ISAKSON and G. J. EISLEY 1964 NASA No. NSG-27-59. Natural frequencies in coupled bending and torsion of twisted rotating and nonrotating blades. 14. S. BANERJEE and J. S. RAO American Society of Mechanical Engineers Paper No. 76-GT.43. Coupled bending-torsion vibrations of rotating blades. 15. W. CARNEGIE, C. STIRLING and J. FLEMING 1966 Paper No. 32. Applied Mechanics Convention, Cambridge, MA. Vibration characteristics of turbine blading under rotation-results of an initial investigation and details of a high-speed test installation. 16. R. M. KRUPKA and A. M. BAUMANIS 1969 Journal of Engineering for Industry American Society of Mechanical Engineers 91, 1017. Bending bending mode of a rotating tapered twisted turbomachine blade including rotary inertia and shear de#ection. 17. J. S. RAO and W. CARNEGIE 1973 International Journal of Mechanical Engineering Education 1, 37}47. Numerical procedure for the determination of the frequencies and mode shapes of lateral vibration of blades allowing for the e!ects of pretwist and rotation. 18. R. MCCALLEY 1963 Report No. DIG SA, 63}73, General Electric Company, Schenectady, New >ork. Rotary inertia correction for mass matrices. 19. K. K. KAPUR 1966 Journal of Acoustical Society of America 40, 1058}1063. Vibrations of a Timoshenko beam, using "nite element approach.
ROTATING TIMOSHENKO BEAMS
117
20. W. CARNEGIE, J. THOMAS and E. DOCUMAKI 1969 Aeronautical Quarterly 20, 321}332. An improved method of matrix displacement analysis in vibration problems. 21. R. NICKEL and G. SECOR 1972 International Journal of Numerical Methods in Engineering 5, 243}253. Convergence of consistently derived Timoshenko beam "nite elements. 22. J. THOMAS and B. A. H. ABBAS 1975 Journal of Sound and
APPENDIX A: EXPRESSIONS FOR STRAIN AND KINETIC ENERGIES A.1. EXPRESSION FOR STRAIN ENERGY (;)
A.1.1. Strain energy due to bending If the bending de#ections in yz and xz planes of a beam are w and v , respectively, the @ @ axial strain and stress induced due to w and v are given by e due to w "!y (*w /*z); @ @ VV @ @ e due to v "!x *v /*z and p due to w "!Ey *w /*z; p due to v " VV @ @ VV @ @ VV @ !Ex *v /*z. @ The strain energy stored in the beam due to bending is given by: ; due to bending 1 " 2
4 eVVpVV d<
1 " 2
4 (eVV due to w@#eVV due to v@) (pVV due to w@#pVV due to v@) d<
*w *v E J @#x @ dA " dz y 2 *z *z
*z@ y dA# *z@ x dA#2 *z@ *z@ xy dA
E J " dz 2
*w
*v
*w *v
2VV *z@ #EIVW *z@ *z@# 2WW *z@ dz,
"
J EI
*w
*w *v
EI
*v
where < is the volume, l is the length and A is the cross-sectional area of the beam.
(A.1)
118
S. S. RAO AND R. S. GUPTA
A.1.2. Strain energy due to shearing Let F and F be the shear forces that produce the shear de#ections dv and dw in an V W Q Q element of length dz respectively. Then the strain energy of the beam due to shearing is given by
1 J dv dw Q#F Q dz. ; " F BSC RM QFC?PGLE 2 V dz W dz By substituting AGk dv /dz and AGk dw /dz for F and F , respectively, one obtains Q Q V W J kAG 2
; " BSC RM QFC?PGLE
dw dv Q # Q dz. dz dz
(A.2)
A.1.3. Strain energy due to rotation The rotation of a beam induces an axial force P in the beam due to centrifugal action. If the beam is bending in the yz plane (Figure A1), the change in the horizontal projection of an element of length ds is given by
ds!dz" (dz)#
*w 1 *w dz !dz+ dz. *z 2 *z
Since the axial force P acts against the changes in the horizontal projection, the work done by P is given by
dw 1 J P(z) dz. ; "! BSC RM . ?LB U 2 dz
(A.3)
The work done by the transverse distributed force p (z) can be written as U J
pU (z) w dz.
; " BSC RM NU
Figure A.1. An element of the beam in equilibrium.
(A.4)
119
ROTATING TIMOSHENKO BEAMS
The expressions corresponding to the bending of the beam in the xz plane can be obtained similarly as
1 J dv ; " P(z) dz, BSC RM . ?LB T 2 dz
J
pT (z) v dz.
" ; BSC RM NT
(A.5, 6)
The total strain energy of the beam can be obtained as given in equation (3) by combining equations (A.1)}(A.6).
A.2.
EXPRESSION FOR KINETIC ENERGY (¹)
Consider a small element of area dA and length dz at a point in the cross-section having co-ordinates (x, y) with respect to x- and y-axes. The kinetic energy of this element is given by o [(wR #vR )#(y Q #x Q )] dz dA, V W 2g where and denote the bending slopes, *v /*z and *w /*z, respectively, and a dot over V W @ @ a symbol represents derivative with respect to time. Integrating this equation over the beam cross-section, the kinetic energy of an element of length dz can be obtained as o d¹" [A(wR #vR )#(I Q #2I Q Q #I Q )] dz. VV W VW V W WW V 2g The kinetic energy of the entire beam (¹ ) can be expressed as
¹"
J oA o (wR #vR )# (I Q #2I Q Q #I Q )] dz, VW V W WW V 2g VV W 2g
(A.7)
which can be seen to be the same as in equation (22).
APPENDIX B: EXPRESSIONS FOR [AK], +BK], 2 , [DM]
The following notation is used for convenience: J
zG\ dz,
;" G
¸ "lG\, i"1, 2, 2 , n, G
zG\ cos (h!h) l #h dz,
<" G
J J
S" G
z
(B.1, 2)
i"1, 2, 2 , n,
(B.3)
z zG\ sin 2 (h !h ) #h dz, i"1, 2, 2 , n, l
(B.4)
where h and h denote the values of pre-twist at nodes 1 and 2, respectively, of the element. As the nature of w , w , v and v is the same except for their positions in the sti!ness and @ Q @ Q mass matrices, one can use wN to denote any one of the quantities w , w , v or v and in @ Q @ Q
120
S. S. RAO AND R. S. GUPTA
TABLE B1
j
1 1 1 1 2 2 2 3 3 4
1 2 3 4 2 3 4 3 4 4
H G H
Q G H I
G H I
k"1
k"2
0 144)0 0 !144)0 1 !72)0 1 !72)0 0 144)0 1 72)0 1 72)0 2 36)0 2 36)0 2 36)0
!144)0 144)0 84)0 60)0 !144)0 !84)0 !60)0 !48)0 !36)0 !24)0
k"3 36)0 !36)0 !24)0 !12)0 36)0 24)0 12)0 16)0 8)0 4)0
k"1 36)0 !36)0 !18)0 !18)0 36)0 18)0 18)0 9)0 9)0 9)0
k"2
k"3
!72)0 72)0 42)0 30)0 !72)0 !42)0 !30)0 !24)0 !18)0 !12)0
36)0 !36)0 !30)0 !12)0 36)0 30)0 12)0 22)0 11)0 4)0
k"4 0)0 0)0 6)0 0)0 0)0 !6)0 0)0 !8)0 !2)0 0)0
k"5 0)0 0)0 0)0 0)0 0)0 0)0 0)0 1)0 0)0 0)0
a similar manner the set (uN , uN , uN , uN ) can be used to represent any one of the sets (u , u , u , u ), (u , u , u , u ), (u , u , u , u ) or (u , u , u , u ). Thus, uN uN uN uN wN (z)" (2z!3lz#l)! (z!2lz#lz)# (3lz!2z)! (z!lz), (B.5) l l l l dwN uN uN uN uN " (6z!6lz)! (3z!4lz#l)# (6lz!6z)! (3z!2lz), dz l l l l dwN uN uN uN uN " (12z!6l)! (6z!4l)# (6l!12z)! (6z!2l). dz l l l l
(B.6)
(B.7)
By letting P (i"1,2, 4; j"1,2, 4; k"1,2, 7) denote the coe$cient of zI\ l\I for G H I the uN uN term in the expression of wN , Q (i"1,2, 4; j"1,2, 4; k"1,2, 5), the G H G H I coe$cient of zI\ l\I for the uN uN term in the expression of (dwN /dz), R (i"1,2, 4; G H G H I j"1,2, 4; k"1,2, 3), the coe$cient of zI\ l\I for the uN uN term in the expression of G H (dwN /dz), H (i"1,2, 4; j"1,2, 4), the index coe$cient of l to account for the G H di!erence in the index of l due to multiplication of rotational degrees of freedom uN and uN and the displacement degrees of freedom uN and uN , the values of P , Q ,R and G H I G H I G H I H can be obtained as shown in Tables B1 and B2. G H B.1.
EVALUATION OF [BK]
As the procedure for the derivation of [AK], [BK],2, [DM] is the same, the expression for [BK] is derived here as an illustration:
EIWW *z@ dz
[u u u u ]2 [BK] [u u u u ]"
"
J
*wN EI dz, WW *z
J
*v
(B.8)
121
ROTATING TIMOSHENKO BEAMS
TABLE B2
j
k"1
1 1 1 1 2 2 2 3 3 4
1 2 3 4 2 3 4 3 4 4
4)0 !4)0 !2)0 !2)0 4)0 2)0 2)0 1)0 1)0 1)0
k"2 !12)0 12)0 7)0 5)0 !12)0 !7)0 !5)0 !4)0 !3)0 !2)0
k"3
k"4
9)0 !9)0 !8)0 !3)0 9)0 8)0 3)0 6)0 3)0 1)0
k"5
4)0 !2)0 2)0 !1)0 0)0 !3)0 0)0 !4)0 !1)0 0)0
k"6 0)0 0)0 !1)0 0)0 0)0 0)0 0)0 0)0 0)0 0)0
!6)0 3)0 2)0 1)0 0)0 0)0 0)0 1)0 0)0 0)0
k"7 1)0 0)0 0)0 0)0 0)0 0)0 0)0 0)0 0)0 0)0
where
wN "v and @
uN u uN u " . uN u uN u
Substituting the values of I and wN into equation (B.8), WW
*wN J z EI dz" E I #(I !I ) cos (h !h ) #h WW *z VYVY WYWY VYVY l
J
uN uN uN uN (12z!6l)! (6z!4l)# (6l!12z)! (6z!2l) dz l l l l
with BK "coe$cient of uN uN "coe$cient of u u
(az#alz#alz#alz#al)
E J " l&G H 12l
#+(d z#d lz#d lz#d lz#d l)
z !(a z#a lz#a lz#a lz#a l), cos (h !h ) #h l
;+R z#R lz#R l, dz
(B.9)
122
S. S. RAO AND R. S. GUPTA
E " ¸ a [R ; #¸ R ; #¸ R < ] & G \G G> \G G> \G > 12l G (d !a ) [¸ R < #¸ R < #¸ R < ] G G G \G G> \G G> \G
(B.10)
E +a ¸ ; R ,#(d !a ) +¸ < R ,. " ¸ G G>H\ \G\H H G G G>H\ \G\H H 12l & > G H This relation can be generalized as E BK " ¸ +a ¸ ; R , ' ( 12l & (> G G>H\ \G\H ' ( H G H #(d !a ) +¸ < R ,, I"1,2, 4, J"I,2, 4, G G G>H\ \G\H ' ( H
(B.11)
E " [+a ¸ ; R , G G>H>&' ( \G\H ' ( H 12l G H < R ,], I"1,2, 4, #(d !a ) +¸ G G G>H>&' ( \G\H ' ( H
J"1,2, 4.
Similarly, E AK " [+d ¸ ; R , ' ( 12l G G>H>&' ( \G\H ' ( H G H
(B.12)
< R ,], I"1,2, 4, J"1,2, 4, #(a !d ) +¸ G G G>H>&' ( \G\H ' ( H kG CK " [c ¸ ; Q ], ' ( l G G>H>&' ( \G\H ' ( H G H
I"1,2, 4,
J"1,2, 4,
E DK " [(a !d )¸ S R ], I"1,2, 4, ' ( 12l G G G>H>&' ( \G\H ' ( H G H o [c ¸ ; P ], I"1,2, 4, AM " G G>H>&' ( \G\H ' ( H ' ( gl G H
J"1,2, 4,
J"1,2, 4,
o BM " [+d ¸ ; Q , ' ( 12gl G G>H>&' ( \G\H ' ( H G H #(a !d ) +¸ < Q ,], G G G>H>&' ( \G\H ' ( H
(B.14) (B.15)
(B.16)
I"1,2, 4, J"1,2, 4,
o CM " [+a ¸ ; Q , ' ( 12gl G G>H>&' ( \G\H ' ( H G H #(d !a ) +¸ < Q ,], G G G>H>&' ( \G\H ' ( H
(B.13)
I"1,2, 4, J"1,2, 4,
(B.17)
123
ROTATING TIMOSHENKO BEAMS
o DM " [(a !d ) ¸ S Q ], ' ( 12gl G G G>H>&' ( \G\H ' ( H G H
(B.18)
I"1,2, 4, J"1,2, 4.
B.2.
EVALUATION OF [EK] AND [FK]
[u u u u ]2 [EK] [u u u u ]" J oAX g
"
P(z)
*w @ dz *z
1 1 e¸# ¸!ez !z !(e#z ) z! z C C C 2 2
1 J oAX e¸# ¸!ez !z " C C 2 g
Thus,
J
*w @ dz *z (B.19)
*w @ dz *z
J oAX *w J 1 oAX *w @ dz! (e#z ) z z ! C g *z 2 g *z@ dz.
1 oX e¸# ¸!ez !z EK " C C ' ( 2 gl
[C ¸ ; Q ] G G>H>&' ( \G\H ' ( H G H
oX ! (e#z ) [C ¸ ; Q ] C G G>H>&' ( \G\H ' ( H gl G H oX ! [C ¸ ; Q ], I"1,2, 4, G G>H>&' ( \G\H ' ( H 2gl G H Similarly,
(B.20) J"1,2, 4.
2oX FK " ; P ], I"1,2, 4, J"I,2, 4. (B.21) [C ¸ ' ( G G>H>&' ( \G\H ' ( H gl G H
APPENDIX C: NOMENCLATURE A b e E g G h I ,I ,I VV WW VW [K] l ¸ [M] t u
area of cross-section breadth of beam o!set Young's modulus acceleration due to gravity shear modulus depth of beam moment of inertia of beam cross-section about xx-, yy- and xy-axis element sti!ness matrix length of an element length of total beam element mass matrix time parameter nodal degrees of freedom
124
S. S. RAO AND R. S. GUPTA
a b h o k X
strain energy displacement in xz plane displacement in yz plane co-ordinate axes co-ordinate axis and length parameter distance of the "rst node of the element from the root of the beam ratio of modal frequency to frequency of fundamental mode of uniform beam with the same root cross-section and without shear deformation e!ects depth taper ratio"h /h breadth taper ratio"b /b angle of twist weight density shear coe$cient rotational speed of the beam (rad/s)
Subscripts b s
bending shear
; v w x, y z z C frequency ratio